Backward Transfer From Quadratic Functions Instruction Onto Different Levels of Development of Conceptions of Linear Function

Author:
Charles Hohensee University of Delaware

Search for other papers by Charles Hohensee in
Current site
Google Scholar
PubMed Close
,
Matthew Melville Purdue University Fort Wayne

Search for other papers by Matthew Melville in
Current site
Google Scholar
PubMed Close
,
Crystal Collier University of Delaware

Search for other papers by Crystal Collier in
Current site
Google Scholar
PubMed Close
, and
Yue Ma University of Delaware

Search for other papers by Yue Ma in
Current site
Google Scholar
PubMed Close
Volume/Issue:
Volume 56: Issue 2
Page(s):
36–62
DOI:
https://doi.org/10.5951/jresematheduc-2022-0187
Restricted access
NCTM members GET ACCESS here
Keywords:
Backward transfer; Linear functions; Quadratic functions; Covariational reasoning

This study examined backward transfer, which we define as how students’ ways of reasoning about previously encountered concepts are modified when learning about new concepts. We examined the backward transfer produced when students learned about quadratic functions. We were specifically interested in how backward transfer may vary for students whose incoming conceptions about linear functions were at different levels of development. Our study comprised a two-week quadratic functions instructional unit emphasizing covariational reasoning bracketed by pre- and postassessments and interviews. Our analysis focused on four students with incoming linear functions conceptions at different levels of development. Findings revealed that students experienced different kinds of backward transfer. This study generated new insights into backward transfer in the context of mathematics education.

Contributor Notes

Charles Hohensee, School of Education, University of Delaware, Newark, DE 19716; hohensee@udel.edu

Matthew Melville, Department of Mathematical Sciences, Purdue University Fort Wayne; Fort Wayne, IN 46805; mdmelvil@pfw.edu

Crystal Collier, School of Education, University of Delaware, Newark, DE 19716; crystalc@udel.edu

Yue Ma, School of Education, University of Delaware, Newark, DE 19716; yuemajoy@udel.edu

  • Collapse
  • Expand
Journal for Research in Mathematics Education
Cover Journal for Research in Mathematics Education

  • Arieli-Attali, M., Wylie, E. C., & Bauer, M. I. (2012, April 13–17). The use of three learning progressions in supporting formative assessment in middle school mathematics [Paper presentation]. Annual meeting of the American Educational Research Association, Vancouver, BC, Canada.

    • Search Google Scholar
    • Export Citation

  • Arzi, H. J., Ben-Zvi, R., & Ganiel, U. (1985). Proactive and retroactive facilitation of long-term retention by curriculum continuity. American Educational Research Journal, 22(3), 369–388. https://doi.org/10.3102/00028312022003369

    • Search Google Scholar
    • Export Citation

  • Bagley, S., Rasmussen, C., & Zandieh, M. (2015). Inverse, composition, and identity: The case of function and linear transformation. The Journal of Mathematical Behavior, 37, 36–47. https://doi.org/10.1016/j.jmathb.2014.11.003

    • Search Google Scholar
    • Export Citation

  • Barab, S., & Squire, K. (2004). Design-based research: Putting a stake in the ground. Journal of the Learning Sciences, 13(1), 1–14. https://doi.org/10.1207/s15327809jls1301_1

    • Search Google Scholar
    • Export Citation

  • Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn? A taxonomy for far transfer. Psychological Bulletin, 128(4), 612–637. https://doi.org/10.1037/0033-2909.128.4.612

    • Search Google Scholar
    • Export Citation

  • Battista, M. T. (2011). Conceptualizations and issues related to learning progressions, learning trajectories, and levels of sophistication. The Mathematics Enthusiast, 8(3), 507–570. https://doi.org/10.54870/1551-3440.1228

    • Search Google Scholar
    • Export Citation

  • Beach, K. (1999). Consequential transitions: A sociocultural expedition beyond transfer in education. Review of Research in Education, 24(1), 101–139. https://doi.org/10.3102/0091732X024001101

    • Search Google Scholar
    • Export Citation

  • Bowers, J., Cobb, P., & McClain, K. (1999). The evolution of mathematical practices: A case study. Cognition and Instruction, 17(1), 25–66. https://doi.org/10.1207/s1532690xci1701_2

    • Search Google Scholar
    • Export Citation

  • Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23(3), 247–285. https://doi.org/10.1007/BF02309532

    • Search Google Scholar
    • Export Citation

  • Bruner, J. (1961). The process of education. Harvard University Press.

  • Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002).Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378. https://doi.org/10.2307/4149958

    • Search Google Scholar
    • Export Citation

  • Cho, P., & Nagle, C. (2017). Procedural and conceptual difficulties with slope: An analysis of students’ mistakes on routine tasks. International Journal of Research in Education and Science, 3(1), 135–150. https://files.eric.ed.gov/fulltext/EJ1126738.pdf

    • Search Google Scholar
    • Export Citation

  • Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2–3), 135–164. https://doi.org/10.1007/BF01273661

    • Search Google Scholar
    • Export Citation

  • Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions. Journal for Research in Mathematics Education, 26(1), 66–86. https://doi.org/10.2307/749228

    • Search Google Scholar
    • Export Citation

  • Cook, V. (Ed.). (2003). Effects of the second language on the first. Multilingual Matters. https://doi.org/10.21832/9781853596346

  • Daugherty, J. L. (2009). Engineering professional development design for secondary school teachers: A multiple case study. Journal of Technology Education, 21(1), 10–24. https://doi.org/10.21061/jte.v21i1.a.1

    • Search Google Scholar
    • Export Citation

  • Detterman, D. K. (1993). The case for the prosecution: Transfer as an epiphenomenon. In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction (pp. 1–24). Ablex.

    • Search Google Scholar
    • Export Citation

  • Doherty-King, B., & Bowers, B. J. (2013). Attributing the responsibility for ambulating patients: A qualitative study. International Journal of Nursing Studies, 50(9), 1240–1246. https://doi.org/10.1016/j.ijnurstu.2013.02.007

    • Search Google Scholar
    • Export Citation

  • Duncker, K. (1945). On problem-solving (L. S. Lees, Trans.). Psychological Monographs, 58(5), i–113. https://doi.org/10.1037/h0093599

  • Ellis, A. B. (2011). Middle school algebra from a functional perspective: A conceptual analysis of quadratic functions. In L. R. Wiest & T. D. Lamberg (Eds.), Proceedings of the 33rd annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 79–87). University of Nevada, Reno.

    • Search Google Scholar
    • Export Citation

  • Fonger, N. L., Ellis, A., B., & Dogan, M. F. (2020). A quadratic growth learning trajectory. The Journal of Mathematical Behavior, 59, Article 100795. https://doi.org/10.1016/j.jmathb.2020.100795

    • Search Google Scholar
    • Export Citation

  • Gick, M. L., & Holyoak, K. J. (1987). The cognitive basis of knowledge transfer. In S. M. Cormier & J. D. Hagman (Eds.), Transfer of learning: Contemporary research and applications (pp. 9–46). Academic Press. https://doi.org/10.1016/B978-0-12-188950-0.50008-4

    • Search Google Scholar
    • Export Citation

  • Ginsburg, H. P. (1997). Entering the child’s mind: The clinical interview in psychological research and practice. Cambridge University Press. https://doi.org/10.1017/CBO9780511527777

    • Search Google Scholar
    • Export Citation

  • Greeno, J. G. (1989). A perspective on thinking. American Psychologist, 44(2), 134–141. https://doi.org/10.1037/0003-066X.44.2.134

  • Greeno, J. G., Smith, D. R., & Moore, J. L. (1993). Transfer of situated learning. In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction (pp. 99–167). Ablex.

    • Search Google Scholar
    • Export Citation

  • Handal, B., & Bobis, J. (2004). Teaching mathematics thematically: Teachers’ perspectives. Mathematics Education Research Journal, 16(1), 3–18. https://doi.org/10.1007/BF03217388

    • Search Google Scholar
    • Export Citation

  • Hohensee, C. (2014). Backward transfer: An investigation of the influence of quadratic functions instruction on students’ prior ways of reasoning about linear functions. Mathematical Thinking and Learning, 16(2), 135–174. https://doi.org/10.1080/10986065.2014.889503

    • Search Google Scholar
    • Export Citation

  • Hohensee, C. (2016). Student noticing in classroom settings: A process underlying influences on prior ways of reasoning. The Journal of Mathematical Behavior, 42, 69–91. https://doi.org/10.1016/j.jmathb.2016.03.002

    • Search Google Scholar
    • Export Citation

  • Hohensee, C., Gartland, S., Melville, M., & Willoughby, L. (2024). Comparing contrasting instructional approaches: A way for research to develop insights about backward transfer. Research in Mathematics Education. Advance online publication. https://doi.org/10.1080/14794802.2024.2388067

    • Search Google Scholar
    • Export Citation

  • Hohensee, C., Gartland, S., Willoughby, L., & Melville, M. (2021). Backward transfer influences from quadratic functions instruction on students’ prior ways of covariational reasoning about linear functions. The Journal of Mathematical Behavior, 61, Article 100834. https://doi.org/10.1016/j.jmathb.2020.100834

    • Search Google Scholar
    • Export Citation

  • Hohensee, C., Willoughby, L., & Gartland, S. (2024). Backward transfer, the relationship between new learning and prior ways of reasoning, and action versus process views of linear functions. Mathematical Thinking and Learning, 26(1), 71–89. https://doi.org/10.1080/10986065.2022.2037043

    • Search Google Scholar
    • Export Citation

  • Karplus, R. (1980). Teaching for the development of reasoning. Research in Science Education, 10(1), 1–9. https://doi.org/10.1007/BF02356303

    • Search Google Scholar
    • Export Citation

  • Kertil, M., Erbas, A. K., & Cetinkaya, B. (2019). Developing prospective teachers’ covariational reasoning through a model development sequence. Mathematical Thinking and Learning, 21(3), 207–233. https://doi.org/10.1080/10986065.2019.1576001

    • Search Google Scholar
    • Export Citation

  • Kurzer, M. (2019). The contrast of covariational reasoning and other problem solving methods of a calculus student [Bachelor’s thesis, Portland State University]. PDX Scholar. https://doi.org/10.15760/honors.603

    • Search Google Scholar
    • Export Citation

  • Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge University Press. https://doi.org/10.1017/CBO9780511609268

    • Search Google Scholar
    • Export Citation

  • Lima, R. N., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18. https://doi.org/10.1007/s10649-007-9086-0

    • Search Google Scholar
    • Export Citation

  • Lobato, J. (2003). How design experiments can inform a rethinking of transfer and vice versa. Educational Researcher, 32(1), 17–20. https://doi.org/10.3102/0013189X032001017

    • Search Google Scholar
    • Export Citation

  • Lobato, J. (2006). Alternative perspectives on the transfer of learning: History, issues, and challenges for future research. Journal of the Learning Sciences, 15(4), 431–449. https://doi.org/10.1207/s15327809jls1504_1

    • Search Google Scholar
    • Export Citation

  • Lobato, J. (2008). Research methods for alternate approaches to transfer: Implications for design experiments. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 167–194). Routledge.

    • Search Google Scholar
    • Export Citation

  • Lobato, J. (2012). The actor-oriented transfer perspective and its contributions to educational research and practice. Educational Psychologist, 47(3), 232–247. https://doi.org/10.1080/00461520.2012.693353

    • Search Google Scholar
    • Export Citation

  • Lobato, J., Ellis, A. B., & Muñoz, R. (2003). How “focusing phenomena” in the instructional environment support individual students’ generalizations. Mathematical Thinking and Learning, 5(1), 1–36. https://doi.org/10.1207/S15327833MTL0501_01

    • Search Google Scholar
    • Export Citation

  • Lobato, J., Hohensee, C., & Rhodehamel, B. (2013). Students’ mathematical noticing. Journal for Research in Mathematics Education, 44(5), 809–850. https://doi.org/10.5951/jresematheduc.44.5.0809

    • Search Google Scholar
    • Export Citation

  • Lobato, J., Hohensee, C., Rhodehamel, B., & Diamond, J. (2012). Using student reasoning to inform the development of conceptual learning goals: The case of quadratic functions. Mathematical Thinking and Learning, 14(2), 85–119. https://doi.org/10.1080/10986065.2012.656362

    • Search Google Scholar
    • Export Citation

  • Lobato, J., & Walters, C. D. (2017). A taxonomy of approaches to learning trajectories and progressions. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 74–101). National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Lubienski, S. T. (2000). Problem solving as a means toward mathematics for all: An exploratory look through a class lens. Journal for Research in Mathematics Education, 31(4), 454–482. https://doi.org/10.2307/749653

    • Search Google Scholar
    • Export Citation

  • MacGregor, M., & Stacey, K. (1997). Students’ understanding of algebraic notation: 11–15. Educational Studies in Mathematics, 33(1), 1–19. https://doi.org/10.1023/A:1002970913563

    • Search Google Scholar
    • Export Citation

  • Moore, K. C. (2021). Graphical shape thinking and transfer. In C. Hohensee & J. Lobato (Eds.), Transfer of learning: Progressive perspectives for mathematics education and related fields (pp. 145–171). Springer. https://doi.org/10.1007/978-3-030-65632-4_7

    • Search Google Scholar
    • Export Citation

  • Moore, K. C., Silverman, J., Paoletti, T., Liss, D., & Musgrave, S. (2019). Conventions, habits, and U.S. teachers’ meanings for graphs. The Journal of Mathematical Behavior, 53, 179–195. https://doi.org/10.1016/j.jmathb.2018.08.002

    • Search Google Scholar
    • Export Citation

  • Moritz, J. (2004). Reasoning about covariation. In D. Ben-Zvi & J. Garfield (Eds.), The challenge of developing statistical literacy, reasoning and thinking (pp. 227–255). Kluwer. https://doi.org/10.1007/1-4020-2278-6_10

    • Search Google Scholar
    • Export Citation

  • Moss, P. A., & Haertel, E. H. (2016). Engaging methodological pluralism. In D. Gitomer & C. Bell (Eds.), Handbook of research on teaching (5th ed., pp. 127–247). American Educational Research Association.

    • Search Google Scholar
    • Export Citation

  • Murata, A. (2008). Mathematics teaching and learning as a mediating process: The case of tape diagrams. Mathematical Thinking and Learning, 10(4), 374–406. https://doi.org/10.1080/10986060802291642

    • Search Google Scholar
    • Export Citation

  • National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common core state standards for mathematics. http://www.corestandards.org

    • Search Google Scholar
    • Export Citation

  • Ogawa, R. T., & Malen, B. (1991). Towards rigor in reviews of multivocal literatures: Applying the exploratory case study method. Review of Educational Research, 61(3), 265–286. https://doi.org/10.3102/00346543061003265

    • Search Google Scholar
    • Export Citation

  • Osgood, C. E. (1949). The similarity paradox in human learning: A resolution. Psychological Review, 56(3), 132–143. https://doi.org/10.1037/h0057488

    • Search Google Scholar
    • Export Citation

  • Peck, F. A. (2020). Beyond rise over run: A learning trajectory for slope. Journal for Research in Mathematics Education, 51(4), 433–467. https://doi.org/10.5951/jresematheduc-2020-0045

    • Search Google Scholar
    • Export Citation

  • Pirie, S., & Kieren, T. (1994). Growth in mathematical understanding: How can we characterise it and how can we represent it? Educational Studies in Mathematics, 26(2–3), 165–190. https://doi.org/10.1007/BF01273662

    • Search Google Scholar
    • Export Citation

  • Reinke, L. T. (2020). Contextual problems as conceptual anchors: An illustrative case. Research in Mathematics Education, 22(1), 3–21. https://doi.org/10.1080/14794802.2019.1618731

    • Search Google Scholar
    • Export Citation

  • Sadler, P., & Sonnert, G. (2018). The path to college calculus: The impact of high school mathematics coursework. Journal for Research in Mathematics Education, 49(3), 292–329. https://doi.org/10.5951/jresematheduc.49.3.0292

    • Search Google Scholar
    • Export Citation

  • Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? The role of computer tools. Journal for Research in Mathematics Education, 30(4), 362–389. https://doi.org/10.2307/749706

    • Search Google Scholar
    • Export Citation

  • Sloane, F. C. (2008). Randomized trials in mathematics education: Recalibrating the proposed high watermark. Educational Researcher, 37(9), 624–630. https://doi.org/10.3102/0013189X08328879

    • Search Google Scholar
    • Export Citation

  • Smith, J. P., III, diSessa, A. A., & Roschelle, J. (1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. Journal of the Learning Sciences, 3(2), 115–163. https://doi.org/10.1207/s15327809jls0302_1

    • Search Google Scholar
    • Export Citation

  • Sosa-Díaz, M.-J., & Valverde-Berrocoso, J. (2022). Grounded theory as a research methodology in educational technology. International Journal of Qualitative Methods, 21, 1–13. https://doi.org/10.1177/16094069221133228

    • Search Google Scholar
    • Export Citation

  • Stacey, K. (1989). Finding and using patterns in linear generalising problems. Educational Studies in Mathematics, 20(2), 147–164. https://doi.org/10.1007/BF00579460

    • Search Google Scholar
    • Export Citation

  • Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory (2nd ed.). Sage.

    • Search Google Scholar
    • Export Citation

  • Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Teachers of Mathematics.

    • Search Google Scholar
    • Export Citation

  • Tuomi-Gröhn, T., & Engeström, Y. (2003). Conceptualizing transfer: From standard notions to developmental perspectives. In T. Tuomi-Gröhn & Y. Engeström (Eds.), Between school and work: New perspectives on transfer and boundary-crossing (pp. 19–38). Emerald.

    • Search Google Scholar
    • Export Citation

  • Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2004). Remedying secondary school students’ illusion of linearity: A teaching experiment aiming at conceptual change. Learning and Instruction, 14(5), 485–501. https://doi.org/10.1016/j.learninstruc.2004.06.019

    • Search Google Scholar
    • Export Citation

  • VanLehn, K., Brown, J. S., & Greeno, J. (1984). Competitive argumentation in computational theories of cognition. In W. Kintsch, J. R. Miller, & P. G. Polson (Eds.), Methods and tactics in cognitive science (pp. 235–262). Psychology Press.

    • Search Google Scholar
    • Export Citation

  • Wertheimer, M. (1945). Productive thinking. Harper.

  • Wilkie, K. J. (2020). Investigating students’ attention to covariation features of their constructed graphs in a figural pattern generalisation context. International Journal of Science and Mathematics Education, 18(2), 315–336. https://doi.org/10.1007/s10763-019-09955-6

    • Search Google Scholar
    • Export Citation

  • Woodworth, R. S., & Thorndike, E. L. (1901). The influence of improvement in one mental function upon the efficiency of other functions. Psychological Review, 8(3), 247–261. https://doi.org/10.1037/h0074898

    • Search Google Scholar
    • Export Citation

  • Young, S. (2015). Investigating backward transfer effects in calculus students. In T. Fukawa-Connely, N. E. Infante, K. Keene, and M. Zandieh (Eds.), Proceedings of the 18th annual conference on Research in Undergraduate Mathematics Education (pp. 1063–1065). RUME.

    • Search Google Scholar
    • Export Citation

  • Yu, F. (2024). Extending the covariation framework: Connecting covariational reasoning to students’ interpretation of rate of change. The Journal of Mathematical Behavior, 73, Article 101122. https://doi.org/10.1016/j.jmathb.2023.101122

    • Search Google Scholar
    • Export Citation
Copyright:
The National Council of Teachers of Mathematics, Inc. All rights reserved. 2025
Page Count:
27
Article Category:
Research Article
Received:
09 Dec 2023
Accepted:
01 Mar 2024
Print Publication Date:
01 Mar 2025
Online Publication Date:
Mar 2025
All Time Past Year Past 30 Days
Abstract Views 56808 56808 47250
Full Text Views 123 123 30
PDF Downloads 152 152 31
EPUB Downloads 0 0 0